spherical surface.

The representation given above of spherical geometry on the
plane is important for us, because it readily allows itself to be
transferred to the three-dimensional case.

Let us imagine a point _S_ of our space, and a great number
of small spheres, _L'_, which can all be brought to coincide with
one another. But these spheres are not to be rigid in the sense
of Euclidean geometry; their radius is to increase (in the sense
of Euclidean geometry) when they are moved away from _S_ towards
infinity, and this increase is to take place in exact accordance
with the same law as applies to the increase of the radii of the
disc-shadows _L'_ on the plane.

After having gained a vivid mental image of the geometrical
behaviour of our _L'_ spheres, let us assume that in our space there
are no rigid bodies at all in the sense of Euclidean geometry, but
only bodies having the behaviour of our _L'_ spheres. Then we shall
have a vivid representation of three-dimensional spherical space,
or, rather of three-dimensional spherical geometry. Here our spheres
must be called "rigid" spheres. Their increase in size as they
depart from _S_ is not to be detected by measuring with
measuring-rods, any more than in the case of the disc-shadows on
_E_, because the standards of measurement behave in the same way as
the spheres. Space is homogeneous, that is to say, the same
spherical configurations are possible in the environment of all
points.* Our space is finite, because, in consequence of the
"growth" of the spheres, only a finite number of them can find room
in space.